Speed: Dimensional Formula Derivation
Speed is defined as the rate at which an object covers distance. Mathematically, it's distance travelled divided by the time taken.
Understanding Dimensions:
- Distance is a measure of length, so its dimension is Length: \([L]\).
- Time is a fundamental quantity, so its dimension is Time: \([T]\).
- Mass is not directly involved in the definition of speed, so its dimension is Mass: \([M^0]\).
Deriving the Formula
The dimensional formula for speed can be derived using its definition:
\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)
Substituting the dimensions:
\([\text{Speed}] = \frac{[L]}{[T]}\)
Using exponent rules, we move \([T]\) to the numerator:
\([\text{Speed}] = [L^1 T^{-1}]\)
Including the dimension for mass (\([M^0]\)), the complete dimensional formula for speed is:
\([\text{Speed}] = [M^0 L^1 T^{-1}]\)
This is often written simply as \([M^0LT^{-1}]\).
Matching with Options
Comparing our derived formula \([M^0LT^{-1}]\) with the given options:
- Option 1: \([ML^0T^{-1}]\)
- Option 2: \([MLT^{-1}]\)
- Option 3: \([M^0LT^{-1}]\)
- Option 4: \([MLT^{-2}]\)
The derived dimensional formula matches Option 3.
Important Questions from Dimensional formulae and dimensional equations
- Considering the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$, where $F$ is force, $q$ is electric charge, and $v$ is velocity, what is the dimensional formula for magnetic flux density $B$?
Given that the energy stored in an inductor is expressed as $U_L = \frac{1}{2}LI^2$ and the power dissipated in a resistor is $P = I^2R$, where $L$ is inductance, $R$ is resistance, and $I$ is current, determine the dimension of the ratio $\frac{L}{R}$.
The dimensions of energy are:
If force $[F]$, acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of pressure.
- The characteristic impedance of free space, $Z_0$, is given by the expression $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}}$. If $\mu_0$ represents the magnetic permeability and $\epsilon_0$ represents the electric permittivity, what are the dimensions of $Z_0$?
